Hypothesis tests

Appendices

Student's t Distribution

The t distribution (aka, Student’s t-distribution)
is a probability distribution that is used to estimate population
parameters when the sample size is small and/or when the population
variance is unknown.

But sample sizes are sometimes small, and often we do not know the
standard deviation of the population.
When either of these problems occur, statisticians rely on the
distribution of the
t statistic (also known as the
t score), whose values are given by:

t = [ x - μ ]
/ [ s / sqrt( n ) ]

where x is the sample mean, μ
is the population mean, s is the standard deviation of the sample, and n is the
sample size. The distribution of the t statistic is called the
t distribution or the
Student t distribution.

The t distribution allows us to conduct statistical analyses on certain data
sets that are not appropriate for analysis, using the normal distribution.

Degrees of Freedom

There are actually many different t distributions. The particular form
of the t distribution is determined by its
degrees of freedom. The degrees of freedom refers
to the number of independent observations in a set of data.

When estimating a mean score or a proportion from a single sample,
the number of independent observations is equal to the sample
size minus one. Hence, the distribution of the t statistic from
samples of size 8 would be described by a t distribution having
8 - 1 or 7 degrees of freedom. Similarly, a t distribution having
15 degrees of freedom would be used with a sample of size 16.

For other applications, the degrees of freedom may be calculated
differently. We will describe those computations as they come up.

Properties of the t Distribution

The t distribution has the following properties:

The mean of the distribution is equal to 0 .

The variance
is equal to v / ( v - 2 ), where v is the
degrees of freedom (see last section) and v> 2.

The variance
is always greater than 1, although it is close to 1 when
there are many degrees of freedom. With infinite degrees of freedom,
the t distribution is the same as the
standard normal distribution.

When to Use the t Distribution

The t distribution can be used with any statistic having a bell-shaped
distribution (i.e., approximately normal). The sampling distribution of a statistic
should be bell-shaped if any of the following
conditions apply.

The population distribution is moderately
skewed,
unimodal, without outliers,
and the sample size is at least 40.

The sample size is greater than 40, without outliers.

The t distribution should not be used with small samples from
populations that are not approximately normal.

Probability and the Student t Distribution

When a sample of size n is drawn from a population having a
normal (or nearly normal) distribution, the sample mean can be
transformed into a t statistic, using the equation presented at the
beginning of this lesson. We repeat that equation below:

t = [ x - μ ]
/ [ s / sqrt( n ) ]

where x is the sample mean, μ
is the population mean, s is the standard deviation of the sample, n is the
sample size, and degrees of freedom are equal to n - 1.

The t statistic produced by this transformation can be associated with
a unique
cumulative probability.
This cumulative probability represents the likelihood of finding
a sample mean less than or equal to x,
given a random sample of size n.

The easiest way to find the probability associated with a particular
t statistic is to use the
T Distribution Calculator,
a free tool provided by Stat Trek.

T Distribution Calculator

The T Distribution Calculator solves common statistics problems, based on the t
distribution. The calculator computes cumulative probabilities, based on simple
inputs. Clear instructions guide you to an accurate solution, quickly and
easily. If anything is unclear, frequently-asked questions and sample problems
provide straightforward explanations. The
calculator is free. It can found in the Stat Trek
main menu under the Stat Tools tab. Or you can tap the button below.

Notation and t Statistics

Statisticians use tα to
represent the t statistic that has a
cumulative probability
of (1 - α).
For example, suppose we were interested in the t statistic having
a cumulative probability
of 0.95. In this example, α would be equal to (1 - 0.95)
or 0.05. We would refer to the t statistic as t0.05

Of course, the value of t0.05 depends on the number
of degrees of freedom. For example,
with 2 degrees of freedom, t0.05 is equal to 2.92;
but with 20 degrees of freedom, t0.05 is equal
to 1.725.

Note: Because the t distribution is symmetric about a mean of zero,
the following is true.

tα = -t1 - alpha
And
t1 - alpha = -tα

Thus, if t0.05 = 2.92, then t0.95 = -2.92.

Test Your Understanding

Problem 1

Acme Corporation manufactures light bulbs. The CEO claims that an average Acme
light bulb lasts 300 days. A researcher randomly selects 15 bulbs for testing.
The sampled bulbs last an average of 290 days, with a standard deviation of 50 days. If
the CEO's claim were true, what is the probability that 15 randomly selected
bulbs would have an average life of no more than 290 days?

Note: There are two ways to solve this problem, using the T Distribution
Calculator. Both approaches are presented below. Solution A is the traditional
approach. It requires you to compute the t statistic, based on data presented in
the problem description. Then, you use the T Distribution Calculator to find
the probability. Solution B is easier. You simply enter the problem data into
the T Distribution Calculator. The calculator computes a t statistic "behind the
scenes", and displays the probability. Both approaches come up with exactly the
same answer.

Solution A

The first thing we need to do is compute the t statistic, based
on the following equation:

where x is the sample mean, μ
is the population mean, s is the standard deviation of the sample, and n is the
sample size.

Now, we are ready to use the T Distribution Calculator.
Since we know the t statistic, we select "T score" from the Random Variable
dropdown box. Then, we enter the following data:

The degrees of freedom are equal to 15 - 1 = 14.

The t statistic is equal to - 0.7745966.

The calculator displays the cumulative probability: 0.226. Hence, if the true
bulb life were 300 days, there is a
22.6% chance that the average bulb life for 15 randomly selected bulbs would
be less than or equal to 290 days.

Solution B:

This time, we will work directly with the raw data from the
problem. We will not compute the t statistic; the T
Distribution Calculator will do that work for us. Since we will work
with the raw data, we select "Sample mean" from the Random Variable dropdown
box. Then, we enter the following data:

The degrees of freedom are equal to 15 - 1 = 14.

Assuming the CEO's claim is true, the population mean equals 300.

The sample mean equals 290.

The standard deviation of the sample is 50.

The calculator displays the cumulative probability: 0.226. Hence, there is a
22.6% chance that the average sampled light bulb will burn out within 290 days.

Problem 2

Suppose scores on an IQ test are normally distributed, with a population mean of 100.
Suppose 20 people are randomly selected and tested. The standard deviation in
the sample group is 15. What is the probability that the average test score in
the sample group will be at most 110?

Solution:

To solve this problem, we will work directly with the raw data
from the problem. We will not compute the t statistic; the T
Distribution Calculator will do that work for us. Since we will work
with the raw data, we select "Sample mean" from the Random Variable dropdown
box. Then, we enter the following data:

The degrees of freedom are equal to 20 - 1 = 19.

The population mean equals 100.

The sample mean equals 110.

The standard deviation of the sample is 15.

We enter these values into the T Distribution Calculator.
The calculator displays the cumulative probability: 0.996. Hence, there is a
99.6% chance that the sample average will be no greater than 110.